Slicing object in several pieces
Please go to the edit mode

Hit the Loop Cut and Slide as shown in the picture below

Hover over on cube and wait for the line appear, move pointer to switch between, horizontal and vertical edges,

scroll your mouse wheel two times to increase the number of cuts to 3, hit enter or double click to finalized it.

You will get your required result.

As shown in the image below:





Here is the solution based on your updates


Add a plane on the top of your cube, slightly bigger than your cube, go to the edit mode and add 3 loop cuts on the plane
Hit A to select all faces of the plane, hit x and delete only face, so that we have only a grid of edges, as shown below:

Select the plane and then your cube (in order), enter edit mode, go to top view (numpad 7), make sure you are in orthographic mode
Hit "knife project" and check the box "Cut Through"





Go to object view and select your cube and edit mode you will get the same loop cuts as you did in your plane, delete the plane for the clean up


I hope it worked for you :)



As per your updates


These edges are not seems to be required.